Searcharxiv⌕ Search

arXiv subjects

Christian Richter

Publications and source records attributed to Christian Richter.

At least 19 recordsLinked to original sources
arXiv2025

Self-affine quadrangles

A quadrangle in the Euclidean plane is called $n$-self-affine if it has a dissection into $n$ affine images of itself. All convex quadrangles are known to be $n$-self-affine for every $n \ge 5$. The only $2$-self-affine convex quadrangles are trapezoids. Here we characterize all $3$-self-affine convex quadrangles, obtaining $5$ one-parameter families and $13$ singular examples of affine types. This way we reduce the quest for all $n$-self-affine convex quadrangles to the open case $n=4$. In addition, we show that there are $n$-self-affine non-convex quadrangles for all $n \ge 3$, but not for $n=2$.

math.CO↗
arXiv2024

Canal Classes and Cheeger Sets

Giannopoulos, Hartzoulaki and Paouris asked in \cite{GHP} whether the best ratio between volume and surface area of convex bodies sharing a given orthogonal projection onto a fixed hyperplane is attained in the limit by a cylinder over the given projection. The answer to the question is known to be negative. In this paper, we prove a characterization of the positive answer in dimension $3$, using the Cheeger set of the common projection. A partial characterization is given in higher dimensions. We also prove that certain canal classes of convex bodies provide families of convex bodies satisfying a closely related inequality for a similar ratio.

math.MG↗
arXiv2024

Self-affinity of discs under glass-cut dissections

A topological disc is called $n$-self-affine if it has a dissection into $n$ affine images of itself. It is called $n$-gc-self-affine if the dissection is obtained by successive glass-cuts, which are cuts along segments splitting one disc into two. For every $n \ge 2$, we characterize all $n$-gc-self-affine discs. All such discs turn out to be either triangles or convex quadrangles. All triangles and trapezoids are $n$-gc-self-affine for every $n$. Non-trapezoidal quadrangles are not $n$-gc-self-affine for even $n$. They are $n$-gc-self-affine for every odd $n \ge 7$, and they are $n$-gc-self-affine for $n=5$ if they aren't affine kites. Only four one-parameter families of quadrangles turn out to be $3$-gc-self-affine. In addition, we show that every convex quadrangle is $n$-self-affine for all $n \ge 5$.

math.CO↗
arXiv2022

Covering rectangles by few monotonous polyominoes

A monotonous polyomino is formed by all lattice unit squares met by the graph of some fixed monotonous continuous function $f:[a,b] \to \mathbb{R}$ with $f(k) \notin \mathbb{Z}$ whenever $k \in \mathbb{Z}$. Our main result says that the least cardinality of a covering of a lattice $(m \times n)$-rectangle by monotonous polyominoes is $\left\lceil \frac{2}{3}\left(m+n-\sqrt{m^2+n^2-mn}\right)\right\rceil$. The paper is motivated by a problem on arrangements of straight lines on chessboards.

math.CO↗
arXiv2021

Coproximinality of linear subspaces in generalized Minkowski spaces

We show that, for vector spaces in which distance measurement is performed using a gauge, the existence of best coapproximations in $1$-codimensional closed linear subspaces implies in dimensions $\geq 2$ that the gauge is a norm, and in dimensions $\geq 3$ that the gauge is even a Hilbert space norm. We also show that coproximinality of all closed subspaces of a fixed dimension implies coproximinality of all subspaces of all lower finite dimensions.

math.MG↗
arXiv2020

The complete characterization of tangram pentagons

The old Chinese puzzle tangram gives rise to serious mathematical problems when one asks for all tangram figures that satisfy particular geometric properties. All $13$ convex tangram figures are known since 1942. They include the only triangular and all six quadrangular tangram figures. The families of all $n$-gonal tangram figures with $n \ge 6$ are either infinite or empty. Here we characterize all $53$ pentagonal tangram figures, including $51$ non-convex pentagons and $31$ pentagons whose vertices are not contained in the same orthogonal lattice.

math.MG↗
arXiv2020

Incongruent equipartitions of the plane

R. Nandakumar asked whether there is a tiling of the plane by pairwise incongruent triangles of equal area and equal perimeter. Recently a negative answer was given by Kupavskii, Pach and Tardos. Still one may ask for weaker versions of the problem, or for the analogue of this problem for quadrangles, pentagons, or hexagons. Several answers were given by the first author in a previous paper. Here we solve three further cases. In particular, our main result shows that there are vertex-to-vertex tilings by pairwise incongruent triangles of unit area and bounded perimeter.

math.CO↗
arXiv2020

Incongruent equipartitions of the plane into quadrangles of equal perimeter

Motivated by a question of R.\ Nandakumar, we show that the Euclidean plane can be dissected into mutually incongruent convex quadrangles of the same area and the same perimeter. As a byproduct we obtain vertex-to-vertex dissections of the plane by mutually incongruent triangles of unit area that are arbitrarily close to the periodic vertex-to-vertex tiling by equilateral triangles.

math.MG↗
arXiv2020

On the monotonicity of the isoperimetric quotient for parallel bodies

The isoperimetric quotient of the whole family of inner and outer parallel bodies of a convex body is shown to be decreasing in the parameter of definition of parallel bodies, along with a characterization of those convex bodies for which that quotient happens to be constant on some interval within its domain. This is obtained relative to arbitrary gauge bodies, having the classical Euclidean setting as a particular case. Similar results are established for different families of Wulff shapes that are closely related to parallel bodies. These give rise to solutions of isoperimetric-type problems. Furthermore, new results on the monotonicity of quotients of other quermassintegrals different from surface area and volume, for the family of parallel bodies, are obtained.

math.MG↗
arXiv2019

Tilings of convex polygons by equilateral triangles of many different sizes

An equilateral triangle cannot be dissected into finitely many mutually incongruent equilateral triangles [Tutte 1948]. Therefore Tuza [Tuza 1991] asked for the largest number $s=s(n)$ such that there is a tiling of an equilateral triangle by $n$ equilateral triangles of $s(n)$ different sizes. We solve that problem completely and consider the analogous questions for dissections of convex $k$-gons into equilateral triangles, $k=4,5,6$. Moreover, we discuss all these questions for the subclass of tilings such that no two tiles are translates of each other.

math.MG↗
arXiv2018

Assessing robustness of radiomic features by image perturbation

Image features need to be robust against differences in positioning, acquisition and segmentation to ensure reproducibility. Radiomic models that only include robust features can be used to analyse new images, whereas models with non-robust features may fail to predict the outcome of interest accurately. Test-retest imaging is recommended to assess robustness, but may not be available for the phenotype of interest. We therefore investigated 18 methods to determine feature robustness based on image perturbations. Test-retest and perturbation robustness were compared for 4032 features that were computed from the gross tumour volume in two cohorts with computed tomography imaging: I) 31 non-small-cell lung cancer (NSCLC) patients; II): 19 head-and-neck squamous cell carcinoma (HNSCC) patients. Robustness was measured using the intraclass correlation coefficient (1,1) (ICC). Features with ICC$\geq0.90$ were considered robust. The NSCLC cohort contained more robust features for test-retest imaging than the HNSCC cohort ($73.5\%$ vs. $34.0\%$). A perturbation chain consisting of noise addition, affine translation, volume growth/shrinkage and supervoxel-based contour randomisation identified the fewest false positive robust features (NSCLC: $3.3\%$; HNSCC: $10.0\%$). Thus, this perturbation chain may be used to assess feature robustness.

cs.CV↗
arXiv2018

A Randomized Multivariate Matrix Pencil Method for Superresolution Microscopy

The matrix pencil method is an eigenvalue based approach for the parameter identification of sparse exponential sums. We derive a reconstruction algorithm for multivariate exponential sums that is based on simultaneous diagonalization. Randomization is used and quantified to reduce the simultaneous diagonalization to the eigendecomposition of a single random matrix. To verify feasibility, the algorithm is applied to synthetic and experimental fluorescence microscopy data.

math.NA↗
arXiv2018

Uniqueness of circumcenters in generalized Minkowski spaces

In an $n$-dimensional normed space every bounded set has a unique circumball if and only if every set of cardinality two has a unique circumball and if and only if the unit ball of the space is strictly convex. When the symmetry of the norm is dropped, i.e., when the centrally symmetric unit ball is replaced by an arbitrary convex body, then the above three conditions are no longer equivalent. We show for the latter case that every bounded set has a unique circumball if and only if every set of cardinality at most $n$ has a unique circumball. We also give an equivalent condition in terms of the geometry of the unit ball. In similar terms we answer the following more general question for every $k \in \{0,\ldots,n-2\}$: When are the dimensions of the sets of all circumcenters of arbitrary bounded sets not larger than $k$?

math.MG↗
arXiv2018

Experimental verification of stopping-power prediction from single- and dual-energy computed tomography in biological tissues

An experimental setup for consecutive measurement of ion and x-ray absorption in tissue or other materials is introduced. With this setup using a 3D-printed sample container, the reference stopping-power ratio (SPR) of materials can be measured with an uncertainty of below 0.1%. A total of 65 porcine and bovine tissue samples were prepared for measurement, comprising five samples each of 13 tissue types representing about 80% of the total body mass (three different muscle and fatty tissues, liver, kidney, brain, heart, blood, lung and bone). Using a standard stoichiometric calibration for single-energy CT (SECT) as well as a state-of-the-art dual-energy CT (DECT) approach, SPR was predicted for all tissues and then compared to the measured reference. With the SECT approach, the SPRs of all tissues were predicted with a mean error of (-0.84 $\pm$ 0.12)% and a mean absolute error of (1.27 $\pm$ 0.12)%. In contrast, the DECT-based SPR predictions were overall consistent with the measured reference with a mean error of (-0.02 $\pm$ 0.15)% and a mean absolute error of (0.10 $\pm$ 0.15)%. Thus, in this study, the potential of DECT to decrease range uncertainty could be confirmed in biological tissue.

physics.med-ph↗
arXiv2017

Methodological accuracy of image-based electron-density assessment using dual-energy computed tomography

Purpose: Electron density is the most important tissue property influencing photon and ion dose distributions in radiotherapy patients. Dual-energy computed tomography (DECT) enables the determination of electron density by combining the information on photon attenuation obtained at two different tube voltages. Most algorithms suggested so far use the CT numbers provided after image reconstruction as input parameters, i.e. are imaged-based. To explore the accuracy that can be achieved with these approaches, we quantify the intrinsic methodological and calibration uncertainty of the seemingly simplest approach.

physics.med-ph↗
arXiv2016

GPU Accelerated Explicit Time Integration Methods for Electro-Quasistatic Fields

Electro-quasistatic field problems involving nonlinear materials are commonly discretized in space using finite elements. In this paper, it is proposed to solve the resulting system of ordinary differential equations by an explicit Runge-Kutta-Chebyshev time-integration scheme. This mitigates the need for Newton-Raphson iterations, as they are necessary within fully implicit time integration schemes. However, the electro-quasistatic system of ordinary differential equations has a Laplace-type mass matrix such that parts of the explicit time-integration scheme remain implicit. An iterative solver with constant preconditioner is shown to efficiently solve the resulting multiple right-hand side problem. This approach allows an efficient parallel implementation on a system featuring multiple graphic processing units.

cs.CE↗